Hub labeling schemes are popular methods for computing distances on road networks and other large complex networks, often answering to a query within a few microseconds for graphs with millions of edges. In this work, we study their algorithmic applications beyond distance queries. We focus on eccentricity queries and distance-sum queries, for several versions of these problems on directed weighted graphs, that is in part motivated by their importance in facility location problems. On the negative side, we show conditional lower bounds for these above problems on unweighted undirected sparse graphs, via standard constructions from "Fine-grained" complexity. However, things take a different turn when the hub labels have a sublogarithmic size. Indeed, given a hub labeling of maximum label size $\leq k$, after pre-processing the labels in total $2^{{O}(k)} \cdot |V|^{1+o(1)}$ time, we can compute both the eccentricity and the distance-sum of any vertex in $2^{{O}(k)} \cdot |V|^{o(1)}$ time. It can also be applied to the fast global computation of some topological indices. Finally, as a by-product of our approach, on any fixed class of unweighted graphs with bounded expansion, we can decide whether the diameter of an $n$-vertex graph in the class is at most $k$ in $f(k) \cdot n^{1+o(1)}$ time, for some "explicit" function $f$.